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ABSTRACT: We consider the relation between higher spin gauge fields and real Kac-Moody
Lie algebras. These algebras are obtained by double and triple extensions of
real forms g_0 of the finite-dimensional simple algebras g arising in
dimensional reductions of gravity and supergravity theories. Besides providing
an exhaustive list of all such algebras, together with their associated
involutions and restricted root diagrams, we are able to prove general
properties of their spectrum of generators with respect to a decomposition of
the triple extension of g_0 under its gravity subalgebra gl(D,R). These results
are then combined with known consistent models of higher spin gauge theory to
prove that all but finitely many generators correspond to non-propagating
fields and there are no higher spin fields contained in the Kac-Moody algebra.
10/2011;
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ABSTRACT: We study possible restrictions on the structure of curvature corrections to gravitational theories in the context of their corresponding Kac--Moody algebras, following the initial work on E10 in Class. Quant. Grav. 22 (2005) 2849. We first emphasize that the leading quantum corrections of M-theory can be naturally interpreted in terms of (non-gravity) fundamental weights of E10. We then heuristically explore the extent to which this remark can be generalized to all over-extended algebras by determining which curvature corrections are compatible with their weight structure, and by comparing these curvature terms with known results on the quantum corrections for the corresponding gravitational theories. Comment: 27 pages
04/2006;
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ABSTRACT: In this paper, two things are done. First, we analyze the compatibility of Dirac fermions with the hidden duality symmetries which appear in the toroidal compactification of gravitational theories down to three spacetime dimensions. We show that the Pauli couplings to the p-forms can be adjusted, for all simple (split) groups, so that the fermions transform in a representation of the maximal compact subgroup of the duality group G in three dimensions. Second, we investigate how the Dirac fermions fit in the conjectured hidden overextended symmetry G++. We show compatibility with this symmetry up to the same level as in the pure bosonic case. We also investigate the BKL behaviour of the Einstein-Dirac-p-form systems and provide a group theoretical interpretation of the Belinskii-Khalatnikov result that the Dirac field removes chaos. Comment: 30 pages
06/2005;
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ABSTRACT: We compute the billiards that emerge in the Belinskii-Khalatnikov-Lifshitz (BKL) limit for all pure supergravities in D=4 spacetime dimensions, as well as for D=4, N=4 supergravities coupled to k (N=4) Maxwell supermultiplets. We find that just as for the cases N=0 and N=8 investigated previously, these billiards can be identified with the fundamental Weyl chambers of hyperbolic Kac-Moody algebras. Hence, the dynamics is chaotic in the BKL limit. A new feature arises, however, which is that the relevant Kac-Moody algebra can be the Lorentzian extension of a twisted affine Kac-Moody algebra, while the N=0 and N=8 cases are untwisted. This occurs for N=5, N=3 and N=2. An understanding of this property is provided by showing that the data relevant for determining the billiards are the restricted root system and the maximal split subalgebra of the finite-dimensional real symmetry algebra characterizing the toroidal reduction to D=3 spacetime dimensions. To summarize: split symmetry controls chaos. Comment: 21 pages
04/2003;
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ABSTRACT: In recent papers, it has been shown that (i) the dynamics of theories involving gravity can be described, in the vicinity of a spacelike singularity, as a billiard motion in a region of hyperbolic space bounded by hyperplanes; and (ii) that the relevant billiard has remarkable symmetry properties in the case of pure gravity in $d+1$ spacetime dimensions, or supergravity theories in 10 or 11 spacetime dimensions, for which it turns out to be the fundamental Weyl chamber of the Kac-Moody algebras $AE_d$, $E_{10}$, $BE_{10}$ or $DE_{10}$ (depending on the model). We analyse in this paper the billiards associated to other theories containing gravity, whose toroidal reduction to three dimensions involves coset models $G/H$ (with $G$ maximally non compact). We show that in each case, the billiard is the fundamental Weyl chamber of the (indefinite) Kac-Moody ``overextension'' (or ``canonical Lorentzian extension'') of the finite-dimensional Lie algebra that appears in the toroidal compactification to 3 spacetime dimensions. A remarkable feature of the billiard properties, however, is that they do not depend on the spacetime dimension in which the theory is analyzed and hence are rather robust, while the symmetry algebra that emerges in the toroidal dimensional reduction is dimension-dependent. Comment: 34 pages, 2 figures
06/2002;
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ABSTRACT: Confirming previous heuristic analyses \`a la Belinskii-Khalatnikov-Lifshitz, it is rigorously proven that certain ``subcritical'' Einstein-matter systems exhibit a monotone, generalized Kasner behaviour in the vicinity of a spacelike singularity. The D-dimensional coupled Einstein-dilaton-p-form system is subcritical if the dilaton couplings of the p-forms belong to some dimension dependent open neighbourhood of zero, while pure gravity is subcritical if D is greater than or equal to 11. Our proof relies, like the recent theorem dealing with the (always subcritical) Einstein-dilaton system, on the use of Fuchsian techniques, which enable one to construct local, analytic solutions to the full set of equations of motion. The solutions constructed are ``general'' in the sense that they depend on the maximal expected number of free functions.
03/2002;
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ABSTRACT: It is shown that the never ending oscillatory behaviour of the generic solution, near a cosmological singularity, of the massless bosonic sector of superstring theory can be described as a billiard motion within a simplex in 9-dimensional hyperbolic space. The Coxeter group of reflections of this billiard is discrete and is the Weyl group of the hyperbolic Kac-Moody algebra E$_{10}$ (for type II) or BE$_{10}$ (for type I or heterotic), which are both arithmetic. These results lead to a proof of the chaotic (``Anosov'') nature of the classical cosmological oscillations, and suggest a ``chaotic quantum billiard'' scenario of vacuum selection in string theory. Comment: 4 pages, 1 figure, minor changes in the text, two references added
12/2000;
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ABSTRACT: Some spatially homogeneous Bianchi type I cosmological models filled with homogeneous ``electric'' $p$-form fields are shown to mimic the never-ending oscillatory behaviour of generic string cosmologies established recently. The validity of the ``Kasner-free-flights plus collisions-on-potential-walls'' picture is also illustrated in the case of known, non-chaotic, superstring solutions. Comment: 12 pages, latex, submitted to Phys. Lett. B
06/2000;
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ABSTRACT: We show that the general solution of the Einstein-Dilaton-antisymmetric-tensors field equations of all superstring theories exhibits a chaotic oscillatory behaviour of the Belinskii-Khalatnikov-Lifshitz type near a cosmological singularity. This result indicates that superstring cosmology is much more complex than is assumed in the scenarios presently discussed in the literature.
General Relativity and Gravitation 01/2000; 32(12):2339-2343. · 2.07 Impact Factor
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ABSTRACT: The complete proof of a theorem announced in [1] on the consistent interactions for (non-chiral) exterior form gauge fields is given. The theorem can be easily generalized to the analysis of anomalies. Its proof amounts to computing the local BRST cohomology H^0(s|d) in the space of local n-forms depending on the fields, the ghosts, the antifields and their derivatives. Comment: 18 pages, no figures, misquotes in references corrected
12/1999;
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ABSTRACT: The general solution of the antifield-independent Wess-Zumino consistency condition is worked out for models involving exterior form gauge fields of arbitrary degree. We consider both the free theory and theories with Chapline-Manton couplings. Our approach relies on solving the full set of descent equations by starting from the last element down (“bottom”).
Nuclear Physics B. 12/1998;
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ABSTRACT: We give the complete list of all first-order consistent interaction vertices for a set of exterior form gauge fields of form degree >1, described in the free limit by the standard Maxwell-like action. A special attention is paid to the interactions that deform the gauge transformations. These are shown to be necessarily of the Noether form "conserved antisymmetric tensor" times "p-form potential" and exist only in particular spacetime dimensions. Conditions for consistency to all orders in the coupling constant are given. For illustrative purposes, the analysis is carried out explicitly for a system of forms with two different degrees p and q (1<p<q<n). Comment: Typos and a minor mistake corrected (below equation (16))
06/1997;
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ABSTRACT: We completely compute the local BRST cohomology $H(s|d)$ of the combined Yang-Mills-2-form system coupled through the Yang-Mills Chern-Simons term ("Chapline-Manton model"). We consider the case of a simple gauge group and explicitely include in the analysis the sources for the BRST variations of the fields ("antifields"). We show that there is an antifield independent representative in each cohomological class of $H(s|d)$ at ghost number 0 or 1. Accordingly, any counterterm may be assumed to preserve the gauge symmetries. Similarly, there is no new candidate anomaly beside those already considered in the literature, even when one takes the antifields into account. We then characterize explicitly all the non-trivial solutions of the Wess-Zumino consistency conditions. In particular, we provide a cohomological interpretation of the Green-Schwarz anomaly cancellation mechanism. Comment: Latex file, no figures, 15 pages
02/1997;
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ABSTRACT: The characteristic cohomology $H^k_{char}(d)$ for an arbitrary set of free $p$-form gauge fields is explicitly worked out in all form degrees $k<n-1$, where $n$ is the spacetime dimension. It is shown that this cohomology is finite-dimensional and completely generated by the forms dual to the field strengths. The gauge invariant characteristic cohomology is also computed. The results are extended to interacting $p$-form gauge theories with gauge invariant interactions. Implications for the BRST cohomology are mentioned. Comment: Latex file, no figures, 44 pages
06/1996;
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ABSTRACT: The general solution of the antifield-independent Wess-Zumino consistency condition is worked out for models involving exterior form gauge fields of arbitrary degree. We consider both the free theory and theories with Chapline-Manton couplings. Our approach relies on solving the full set of descent equations by starting from the last element down ("bottom"). * henneaux@ulb.ac.be, bknaepen@ulb.ac.be 1 Introduction p-form gauge theories are generalizations of electromagnetism in which the vector potential -- a 1-form -- is replaced by exterior forms of higher degree. They play an important role in supergravity and superstring theory. The purpose of this paper is to derive the general solution of the antifieldindependent Wess-Zumino consistency condition [1] for free p-form theories -- and theories with interactions that do not deform the gauge symmetry of the free case -- as well as for theories involving Chapline-Manton couplings [2]. As is well known, the Wess-Zumino consisten...
02/1970;
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ABSTRACT: It is shown in detail that the dynamics in the vicinity of a spacelike singularity of the D-dimentional Einstein-dilaton-p-form system can be described, at each spatial point, as a billiard motion in a region of hyperbolic space. This is done within the Hamiltonian formalism. A key role is played in the derivation by the Iwasawa decomposition of the spatial metric. We also comment on the strong coupling/small tension limit of the theory.
Proceedings of School on Quantum Gravity, Kluwer Academic / Plenum Press (2002).
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ABSTRACT: Some time ago, it was found that the never-ending oscillatory chaotic behaviour discovered by Belinskii, Khalatnikov and Lifshitz (BKL) for the generic solution of the vacuum Einstein equations in the vicinity of a spacelike (“cosmological”) singularity disappears in spacetime dimensions D≡d+1>10. Recently, a study of the generalization of the BKL chaotic behaviour to the superstring effective Lagrangians has revealed that this chaos is rooted in the structure of the fundamental Weyl chamber of some underlying hyperbolic Kac–Moody algebra. In this Letter we show that the same connection applies to pure gravity in any spacetime dimension ⩾4, where the relevant algebras are AEd. In this way the disappearance of chaos in pure gravity models in D⩾11 dimensions becomes linked to the fact that the Kac–Moody algebras AEd are no longer hyperbolic for d⩾10.
Physics Letters B.
